A000205 Number of positive integers <= 2^n of form x^2 + 3 y^2.
1, 1, 3, 4, 8, 14, 25, 45, 82, 151, 282, 531, 1003, 1907, 3645, 6993, 13456, 25978, 50248, 97446, 189291, 368338, 717804, 1400699, 2736534, 5352182, 10478044, 20531668, 40264582, 79022464, 155196838, 304997408, 599752463, 1180027022, 2322950591, 4575114295
Offset: 0
Keywords
References
- N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Links
- Seth A. Troisi, Table of n, a(n) for n = 0..45
- P. Moree and H. J. J. te Riele, The hexagonal versus the square lattice, arXiv:math/0204332 [math.NT], 2002.
- P. Moree and H. J. J. te Riele, The hexagonal versus the square lattice, Math. Comp. 73 (2004), no. 245, 451-473.
- D. Shanks and L. P. Schmid, Variations on a theorem of Landau. Part I, Math. Comp., 20 (1966), 551-569.
- Index entries for sequences related to populations of quadratic forms
Programs
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Mathematica
(* This program is not suitable to compute more than a score of terms. *) a[0] = a[1] = 1; a[n_] := a[n] = Module[{cnt, k, r, x, y}, For[cnt = a[n-1]; k = 2^(n-1)+1, k <= 2^n, k++, r = Reduce[x >= 0 && y >= 0 && k == x^2 + 3 y^2, {x, y}, Integers]; If[r =!= False, cnt++]]; cnt]; Table[Print[n, " ", a[n]]; a[n], {n, 0, 20}] (* Jean-François Alcover, Jan 23 2019 *)